Theorem wcomdr 421

Description: Commutation dual. Kalmbach 83 p. 23. (Contributed by NM, 13-Oct-1997.)

Hypothesis

Ref	Expression
wcomdr.1	(a ≡ ((a ∪ b) ∩ (a ∪ b⊥ ))) = 1

Assertion

Ref	Expression
wcomdr	C (a, b) = 1

Proof of Theorem wcomdr

Step	Hyp	Ref	Expression
1		wcomdr.1	. . . . 5 (a ≡ ((a ∪ b) ∩ (a ∪ b⊥ ))) = 1
2		df-a 40	. . . . . . 7 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥
3	2	bi1 118	. . . . . 6 (((a ∪ b) ∩ (a ∪ b⊥ )) ≡ ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥ ) = 1
4		oran 87	. . . . . . . . . 10 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
5	4	bi1 118	. . . . . . . . 9 ((a ∪ b) ≡ (a⊥ ∩ b⊥ )⊥ ) = 1
6	5	wcon2 208	. . . . . . . 8 ((a ∪ b)⊥ ≡ (a⊥ ∩ b⊥ )) = 1
7		oran 87	. . . . . . . . . 10 (a ∪ b⊥ ) = (a⊥ ∩ b⊥ ⊥ )⊥
8	7	bi1 118	. . . . . . . . 9 ((a ∪ b⊥ ) ≡ (a⊥ ∩ b⊥ ⊥ )⊥ ) = 1
9	8	wcon2 208	. . . . . . . 8 ((a ∪ b⊥ )⊥ ≡ (a⊥ ∩ b⊥ ⊥ )) = 1
10	6, 9	w2or 372	. . . . . . 7 (((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))) = 1
11	10	wr4 199	. . . . . 6 (((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥ ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ ) = 1
12	3, 11	wr2 371	. . . . 5 (((a ∪ b) ∩ (a ∪ b⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ ) = 1
13	1, 12	wr2 371	. . . 4 (a ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ ) = 1
14	13	wcon2 208	. . 3 (a⊥ ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))) = 1
15	14	wdf-c1 383	. 2 C (a⊥ , b⊥ ) = 1
16	15	wcomcom5 420	1 C (a, b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by: (None)